91Ó°ÊÓ

In probability theory, the Probability Addition Rule is a fundamental concept used to determine the likelihood of either one event or another occurring. It is helpful in finding the union of two probabilities. The rule can be expressed as: \[P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Here's a quick breakdown of the formula:

• \(P(A \cup B)\): Probability that event \(A\) or event \(B\) (or both) occur.
• \(P(A)\): Probability that event \(A\) occurs.
• \(P(B)\): Probability that event \(B\) occurs.
• \(P(A \cap B)\): Probability that both events \(A\) and \(B\) occur.

For mutually exclusive events, \(P(A \cap B) = 0\), simplifying the equation to: \[P(A \cup B) = P(A) + P(B) \] This rule's simplicity is very useful as it directly allows us to verify if events can indeed be mutually exclusive, as demonstrated in the exercise.

Probability is a measure of the likelihood that a particular event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. Here's a quick look at how probabilities are generally represented:

• \(0 \leq P(E) \leq 1\), where \(E\) is any event.
• A probability of 0 means the event cannot happen.
• A probability of 1 means the event is certain to happen.
• Probabilities in between 0 and 1 describe varying degrees of likelihood.

In the problem presented, the probabilities \(P(A)=0.4\) and \(P(B)=0.7\) were analyzed. Probabilities must sum to no more than 1 for mutually exclusive events. If their sum exceeds this, like in the case when \(P(A) + P(B) = 1.1\), it indicates that the events cannot be mutually exclusive.

Calculating the probability of an event involves using specific principles, depending on the nature of the events involved. For individual events, the basic probability provided can often suffice. However, calculating the probability of compound events—such as the union of two or more events—often requires applying rules like the Probability Addition Rule.
Mutually exclusive events are a special scenario where the joint probability of both events occurring is zero. To understand this better, consider the example where \(P(A) = 0.4\) and \(P(B) = 0.3\). With mutually exclusive events, their combined probability is computed directly as: \[P(A \cup B) = P(A) + P(B) = 0.4 + 0.3 = 0.7 \] Such calculations are straightforward, provided the sum does not exceed 1. This guides us in determining whether two events can feasibly be mutually exclusive. The ease of calculation is why understanding these probabilities and their rules is essential for proper interpretation.